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It is known that some particular self-mappings of the closed unit ball Bl2 of l2 with no fixed points cannot be nonexpansive with respect to any renorming of l2. We give here a short proof of this ...

We give conditions under which a self-mapping on a bounded closed convex subset, containing zero, of a reflexive Banach space is never nonexpansive, i.e., there is no renorming with respect to which the mapping is nonexpansive. This provides with a unifying procedure to prove that the classical mappings of Kakutani, Nirenberg and P. K. Lin are not uniformly Lipschitzian in any bounded closed convex subset of $$\ell _2$$ which they leave invariant. In an analogous way, the well known fixed point free mappings of Goebel-Kirk-Thele and Baillon, although they are uniformly Lipschitzian, are also seen to be never nonexpansive in any subset of $$\ell _2$$ where they are self-mappings.

We find lower bounds for the set of Lipschitz constants of a given Lipschitzian map, defined on the closed unit ball of a Hilbert space, with respect to any renorming. We introduce a class of maps, defined in the closed unit ball of l 2 , which contains the classical fixed point free maps due to Goebel–Kirk–Thelle, Baillon, and P.K. Lin. We show that for any map of this class its uniform Lipschitz constant with respect to any renorming of l 2 is never strictly less than π 2 .

In this paper we prove some norm inequalities in the classical James sequence space J , and use them to prove that orthogonal convexity is a stable property in J .

In this paper, we present a further study of iterated nonexpansive mappings, that is, mappings which are nonexpansive along the orbits. This is a wide class of nonlinear mappings including many generalized nonexpansive mappings, such as Suzuki (C)-type generalized nonexpansive mappings and, among others, mappings satisfying the so-called condition (B). In some cases, as for Suzuki (C)-type generalized nonexpansive mappings, the existence of a fixed point is known in the setting of Banach spaces with normal structure. We prove that the same is true for many other classes of iterated nonexpansive mappings.

We define a class of nonlinear mappings which is properly larger than the class of nonexpansive mappings. We also give a fixed point theorem for this new class of mappings.

In this paper, we introduce a Halpern’s type method to approximate common fixed points of a nonexpansive mapping T and a strongly quasi-nonexpansive mappings S, defined in a Hilbert space, such that I − S is demiclosed at 0. The result shows as the same algorithm converges to different points, depending on the assumptions of the coefficients. Moreover, a numerical example of our iterative scheme is given.

We study some properties of O -convex and P -convex spaces and give a new proof of the P ( n ) -convexity of l p sums of P ( n ) -convex spaces. Further we show using Ramsey’s theorem that the space E β is P -convex. We define some moduli that characterize O -convexity and P -convexity and compare them to a modulus defined by Garcia Falset. This enables us to establish some fixed point theorems. By means of several examples, we separate P -convexity from several geometrical conditions known to imply the fixed point property (FPP).

A very general class of multivalued generalized nonexpansive mappings is defined. We also give some fixed point results for these mappings, and finally we compare and separate this class from the other multivalued generalized nonexpansive mappings introduced in the recent literature.

In this paper we introduce two new classes of generalized nonexpansive mapping and we study both the existence of fixed points and their asymptotic behavior.

We give some fixed point results for pseudocontractive mappings on nonbounded domains which allow us to obtain generalizations of recent fixed point theorems of Penot, Isac, and Németh. An application to integral equations is given.

In this paper we prove the existence of a fixed point for several classes of mappings (mappings admitting a center, nonexpansive mappings, asymptotically nonexpansive mappings) defined on the closed convex subsets of a Banach space satisfying some proximinality conditions. In particular, we derive a sufficient condition, more general than weak star compactness, such that if C is a bounded closed convex subset of l 1 satisfying this condition, then every nonexpansive mapping T : C → C has a fixed point.

Fractals and multivalued fractals play an important role in biology, quantum mechanics, computer graphics, dynamical systems, astronomy and astrophysics, geophysics, etc. Especially, there are important consequences of the iterated function (or multifunction) systems in several topics of applied sciences [see for example: El Naschie MS. Iterated function systems and the two-slit experiment of quantum mechanics. Chaos, Solitons & Fractals 1994;4:1965–8; Iovane G. Cantorian spacetime and Hilbert space: Part I-Foundations. Chaos, Solitons & Fractals 2006;28:857–78; Iovane G. Cantorian space-time and Hilbert space: Part II-Relevant consequences. Chaos, Solitons & Fractals 2006;29:1–22; Fedeli A. On chaotic set-valued discrete dynamical systems. Chaos, Solitons & Fractals 2005;23:13814; Shi Y, Chen G. Chaos of discrete dynamical systems in complete metric spaces. Chaos, Solitons & Fractals 2004;22:55571]. The purpose of this paper is twofold. First, some existence and uniqueness results for the self-similar sets of a mixed iterated function systems are given. Then, using the concept of well-posed fixed point problem, the well-posedness of the self-similarity problem for some classes of iterated multifunction systems is also studied. Well-posedness is closely related to the approximation of the solution of a fixed point equation, which is an important aspect of the construction of the fractals using the so-called pre-fractals.

In this paper we show that the well-known Monch fixed point theorem for non-self mappings remains valid if we replace the Leray–Schauder boundary condition by the interior condition. As a consequence, we obtain a partial generalization of Petryshyn's result for nonexpansive mappings.

We consider the triangles with vertices x, −x and y where x,y are points on the unit sphere of a normed space. Using the geometric means of the variable lengths of the sides of these triangles, we define two geometric constants for Banach spaces. These constants are closely related to the modulus of convexity of the space under consideration, and they seem to represent a useful tool to estimate the exact values of the James and Jordan–von Neumann constants of some Banach spaces.

We introduce a class of nonlinear continuous mappings in Banach spaces which allow us to characterize the Banach spaces without noncompact flat parts in their spheres as those that have the fixed point property for this type of mapping. Later on, we give an application to the existence of zeroes for certain kinds of accretive operators.

It is shown that if the modulus Γ X of nearly uniform smoothness of a reflexive Banach space satisfies Γ X ′ ( 0 ) 1 , then every bounded closed convex subset of X has the fixed point property for nonexpansive mappings. In particular, uniformly nonsquare Banach spaces have this property since they are properly included in this class of spaces. This answers a long-standing question in the theory.

We give some sufficient conditions for normal structure in terms of the von Neumann-Jordan constant, the James constant and the weak orthogonality coefficient introduced by B. Sims. In the rest of the paper, the von Neumann-Jordan constant and the James constant for the Bynum space ℓ 2 , ∞ \ell _{2,\infty } are computed, and are used to show that our results are sharp.

We consider a family of spaces wider than UNC spaces introduced by Prus, and we give some fixed point results in the setting of these spaces.

We give an example of a renorming ofℓ2with the fixed-point property (FPP) for nonexpansive mappings, but which seems to fall out of the scope of all the commonly known sufficient conditions for FPP.

In 2005 Prus and Sczcepanik introduced a large class of Banach spaces with the fixed point property for nonexpansive mappings. We say that this class satisfies the PSz condition. Checking that a given Banach space belongs to this class is not an easy task. Here we study the relationship between the PSz condition and other well-known geometrical properties of Banach spaces, and we give easier sufficient conditions for a Banach space to satisfy it. Consejo Nacional de Ciencia y Tecnologia (México) Ministerio de Economía y Competitividad

It is shown that two well-known uniformly fixed point free lipschitzian semigroups of mappings have minimal Lipschitz constant on the positive part of the unit ball of ℓ 2 \ell _2 . This implies that a question raised by T. Kuczumow has a negative answer.

A closed convex subset of c0 has the fixed point property (fpp) if every nonexpansive self mapping of it has a fixed point. All nonempty weak compact convex subsets of c0 are known to have the fpp. We show that closed convex subsets with a nonempty interior and nonempty convex subsets which are compact in a topology slightly coarser than the weak topology may fail to have the fpp.

We show that the l1 sum of the van Dulst space with itself possesses the fixed point property, although it fails most of the known conditions that imply (FPP). Mathematics Subject Classification: Primary, 46B20; 47H09.

We study some aspects of the fixed point theory for a class of generalized nonexpansive mappings, which among others contain the class of generalized nonexpansive mappings recently defined by Suzuki in 2008.

In this note we give a sufficient condition for the fixed point property for nonexpansive mappings in Banach spaces, which we call the AMCP. This property is more general than normal structure. and the space c 0 , James space, as well as every Banach space endowed with a suitable Schauder basis also have the AMCP. On the other hand, a renorming of the Tsirelson space and also 1 ∞ lack the AMCP.

Indeed, there are a lot of quantitative descriptions of geometrical properties of Banach spaces. The most common way for creating these descriptions, is to define a real function (a “modulus” depending on the Banach space under consideration, and from this define a suitable constant or coefficient closely related to this function. The moduli and/or the constants are attempts to get a better understanding about two things: The shape of the unit ball of a space, and The hidden relations between weak and strong convergence of sequences.

In 1971 Zidler [Zi 71] showed that every separable Banach space (X, ‖·‖) admits an equivalent renorming, (X, ‖·‖0), which is uniformly convex in every direction (UCED), and consequently it has weak normal structure and so the weak fixed point property (WFPP) [D-J-S 71].

In this paper we exhibit some connections between the Dunkl–Williams constant and some other well-known constants and notions. We establish bounds for the Dunkl–Williams constant that explain and quantify a characterization of uniformly nonsquare Banach spaces in terms of the Dunkl–Williams constant given by M. Baronti and P.L. Papini. We also study the relationship between Dunkl–Williams constant, the fixed point property for nonexpansive mappings and normal structure.

Let A be the set of all equivalent norms on l 1 which satisfy the FPP. We prove that A contains rays. In fact, every renorming in l 1 which verifies condition (⁎) in Theorem 2.1 is the starting point of a (closed or open) ray composed by equivalent norms on l 1 with the FPP. The standard norm ‖ ⋅ ‖ 1 or P.K. Linʼs norm defined in Lin (2008) [12] are examples of such norms. Moreover, we study some topological properties of the set A with respect to some equivalent metrics defined on the set of all norms on l 1 equivalent to ‖ ⋅ ‖ 1 .

A famous open question in metric Fixed Point Theory is whether every Banach space which is isomorphic to the Hilbert space l2 has the fixed point property for nonexpansive mappings. We give an overview about the state of the advances towards their solution. Open image in new window

Traditionally, metric fixed point theory has sought classes of spaces in which a given type of mapping (nonexpansive, assymptotically or generalized nonexpansive, uniformly Lipschitz, etc.) from a nonempty weakly compact convex set into itself always has a fixed point. In some situations the class of space is determined by the application while there is some degree of freedom in constructing the map to be used. With this in mind we seek to relax the conditions on the space by considering more restrictive types of mappings.

We give a sufficient condition for normal structure more general than the well known ɛ0(X) < 1. Moreover we obtain sufficient conditions for the fixed point property for some B-convex Banach spaces.

In this paper we present a fixed point theorem for contractive type multivalued operators in cone metric spaces by using the concept of c-distance.